Barycenters and a law of large numbers in Gromov hyperbolic spaces


We investigate barycenters of probability measures on Gromov hyperbolic spaces, toward development of convex optimization in this class of metric spaces. We establish a contraction property in terms of the Wasserstein distance, a deterministic approximation of barycenters of uniform distributions on finite points, and a kind of law of large numbers. These generalize the corresponding results on CAT(0)-spaces, up to additional terms depending on the hyperbolicity constant.